reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve r,s for Real;
reserve p,p1,p2,p3 for Prime;
reserve z for Complex;

theorem Th54:
  for a,b,c,d being positive Real st a/b < 1 & c/d < 1 holds (a/b) * (c/d) < 1
  proof
    let a,b,c,d be positive Real;
    assume a/b < 1;
    then
A1: a < b by XREAL_1:181;
    assume c/d < 1;
    then c < d by XREAL_1:181;
    then
A2: a*c < b*d by A1,XREAL_1:96;
    (a/b) * (c/d) = (a*c) / (b*d) by XCMPLX_1:76;
    hence (a/b) * (c/d) < 1 by A2,XREAL_1:189;
  end;
